The Binary Mass Functions

The masses of the two stars can be described by the X-ray and optical mass func- tions of the binary system:

fX(M) = Kx3P(1 − e2)3/2 2πG = Mcsin3i (1 + q)2 (5.1) and fC(M) = Kc3P(1 − e2)3/2 2πG = Mxsin3i (1 + 1/q)2 (5.2)

respectively. Mx and Mc are the masses of the neutron star and the optical coun-

terpart, i is the inclination of the orbital plane to the line of sight, P is the orbital period, e is the eccentricity and q is the mass ratio (=Mx/Mc). The semiamplitudes

of the radial velocity curves are given by

Kn =

2πansin i

Table 5.6: The derived X-ray mass function, orbital inclination and semimajor axis for the 4 systems studied in this chapter and the 2 previously studied SMC BeXRBs. Mcis estimated from the spectral classification of the authors referenced

here using the table of luminosity and spectral type from de Jager & Nieuwen- huijzen (1987), along with the standard mass-luminosity relationship for main se- quence stars (L∝M3.6). fX(M) is calculated using results from the orbital fits and

is shown in solar masses. The inclination and semimajor axis are estimated using equation 5.1, Mcand Mx= 1.4M⊙. Source (Mc/M⊙) ( fX(M)/M⊙) i(◦) ax(R⋆) SXP2.371 18–26 1.27± 0.05 24+2₋₃ 9 SXP6.851 13–26 7.71± 0.92 50+20₋₉ 12 SXP8.801 13–23 1.86± 0.25 30+6₋₄ 13 SXP74.71 6–9 3.06± 0.94 55+20₋₁₃ 16 SXP11.52 13–26 3.79± 0.48 37+11₋₆ 16 SXP18.33 8–13 1.43± 0.17 36+5₋₇ 10

Note. References for the spectral classification are given: 1McBride et al. (2008),

2_{Chapter 4 of this work and Townsend et al. (2011),}3_{Schurch et al. (2009). The}

spectral type of SXP18.3 in Schurch et al. (2009) is estimated based on optical and NIR photometry as no spectrum allowing for classification is available.

where ax and ac are the semimajor axes of the ellipse travelled by the neutron star

and the optical counterpart about the centre of mass of the system. Thus, if the radial velocity curves for both stars are known, along with the inclination angle, one can calculate the masses of the two stars to high precision. However, due to the non-eclipsing nature of these SMC binary systems, I am unable to make precise measurements of the neutron star masses as the inclination is not well constrained. Instead, the X-ray mass function for each system can be calculated using their or- bital solutions. From this, estimates of the inclination and orbital size (≃ ax) can be

made using masses estimated from the spectral classification of the counterpart and the standard mass of a neutron star of 1.4 M⊙. These results are shown in Table 5.6 for the 6 SMC BeXRBs with orbital solutions. The range in the value of each incli- nation is based mostly on an uncertainty in the spectral classification of one spectral type either side of the published value. The uncertainty in the mass function also contributes to the range in inclination, albeit less significantly. The semimajor axes are estimated based on the most probable mass and radius of the primary star.

The orbital inclinations seen in this sample are as expected given the method of detection. Very low inclinations would mean the delays in pulse arrival times would

not be detected, whilst very high inclinations would mean the X-ray source gets eclipsed by the primary star. More specifically, one can compare the inclinations to Hα profiles of the Be star to investigate the (mis-)alignment of the orbital plane and the circumstellar disk. I find that SXP6.85 and SXP74.7 have quite narrow, single peaked Hα emission (see Chapter 3), despite having the highest estimated inclinations. Conversely, SXP18.3 has a lower inclination but shows prominent double peaked Hαemission in its optical spectrum. However, I note that the mass of SXP18.3 has been estimated photometrically, not spectroscopically. Although this may provide some qualitative evidence of orbit-disk misalignment in these systems, more quantitative evidence may only be obtained from detailed polarimetric studies of the disk itself.

The estimated semimajor axes in Table 5.6 are quite similar and range from 9 to 16 stellar radii, but are these values what one might expect? The use of complex models that describe the dynamics and thermal structure of the circumstellar disk around the primary star helps explain what is observed here. Okazaki (2007) shows that the density of the disk is several orders of magnitude lower beyond 10 R⋆than it is closer to the star, whilst Carciofi (2010) shows most of the optical and NIR flux and polarisation is emitted from within 10 R⋆ and nearly all FIR and Hα flux is produced inside 20 R⋆ (see Chapter 1). Thus, these models are predicting that a very large fraction of the matter in the disk is within approximately 10 R⋆. This prediction goes some way to explaining the nature of the X-ray outbursts that are seen in the SMC binary systems. SXP74.7 and SXP11.5 have the largest predicted orbital size (relative to the radius of the counterpart) and are observationally the least active of the sample, rarely undergoing a Type I outburst and only once being seen in a Type II outburst. SXP6.85, SXP8.80 and SXP18.3 have smaller predicted orbital sizes and are much more active, often being detected in Type I or Type II outbursts, or outbursts in-between these classical types as discussed in Chapter 2. This is likely due to the neutron star passing closer to, or further into, the circumstellar material than those in larger orbits. The exception to this seems to be SXP2.37 which has the smallest predicted orbital size, but is very rarely detected in X-ray outburst. This can be explained when considering the low eccentricity of the orbit. Okazaki & Negueruela (2001) predict that for low eccentricity HMXBs, the circumstellar disk will get truncated at the 3:1 resonance radius, in contrast to intermediate and high eccentricity systems in which truncation is much less efficient and occurs at larger distances near the Roche lobe radius of the star. Thus, the disk in SXP2.37 could be truncated at a much smaller radius than those of the other systems in the sample, explaining the small number of X-ray outbursts seen.

1 10 100 Orbital Period (d) Spin Period (s) 10−2 10−1 100 101 102 103 104

Figure 5.5: Corbet Diagram for all the HMXBs in Table 5.7 that have a known spin period (excluding the PSR systems). The red diamonds represent the SG systems, the blue triangles are the Galactic BeXRBs and the green stars are the SMC BeXRBs.